LBan-1

Overview

This lesson is structured in three parts:

• Self-averaging and disorder in statistical systems

Disordered systems are characterized by a random energy landscape, however, in the thermodynamic limit, physical observables become deterministic. This property, known as self-averaging, does not always hold for the partition function which is the quantity that we can compute. When it holds the annealed average $\ln {\overline {Z}}$ and the quenched average $\ln Z$ coincides otherwiese we have

$\ln Z<\ln {\overline {Z}}$

• The Random Energy Model

We study the Random Energy Model (REM) introduced by Bernard Derrida. In this model at each configuration is assigned an independent energy drawn from a Gaussian distribution of extensive variance. The model exhibits a freezing transition at a critical temperature​, below which the free energy becomes dominated by the lowest energy states.

• Extreme value statistics and saddle-point analysis

The results obtained from a saddle-point approximation can be recovered using the tools of extreme value statistics.

Part I

Random energy landascape

In a system with $N$ degrees of freedom, the number of configurations grows exponentially with $N$. For simplicity, consider Ising spins that take two values, $\sigma _{i}=\pm 1$, located on a lattice of size $L$ in $d$ dimensions. In this case, $N=L^{d}$ and the number of configurations is $M=2^{N}=e^{N\log 2}$.

In the presence of disorder, the energy associated with a given configuration becomes a random quantity. For instance, in the Edwards-Anderson model:

$E=-\sum _{\langle i,j\rangle }J_{ij}\sigma _{i}\sigma _{j},$

where the sum runs over nearest neighbors $\langle i,j\rangle$, and the couplings $J_{ij}$ are independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and unit variance.

The energy of a given configuration is a random quantity because each system corresponds to a different realization of the disorder. In an experiment, this means that each of us has a different physical sample; in a numerical simulation, it means that each of us has generated a different set of couplings $J_{ij}$.

To illustrate this, consider a single configuration, for example the one where all spins are up. The energy of this configuration is given by the sum of all the couplings between neighboring spins:

$E[\sigma _{1}=1,\sigma _{2}=1,\ldots ]=-\sum _{\langle i,j\rangle }J_{ij}.$

Since the the couplings are random, the energy associated with this particular configuration is itself a Gaussian random variable, with zero mean and a variance proportional to the number of terms in the sum — that is, of order $N$. The same reasoning applies to each of the $M=2^{N}$ configurations. So, in a disordered system, the entire energy landscape is random and sample-dependent.

Self-averaging observables

A crucial question is whether the macroscopic properties measured on a given sample are themselves random or not. Our everyday experience suggests that they are not: materials like glass, ceramics, or bronze have well-defined, reproducible physical properties that can be reliably controlled for industrial applications.

From a more mathematical point of view, it means that the free energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_N(\beta)=N f_N(\beta)} and its derivatives (magnetization, specific heat, susceptibility, etc.), in the limit $N\to \infty$, these random quantities concentrates around a well defined value. These observables are called self-averaging. This means that,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N \to \infty} f_N (\beta)= \lim_{N \to \infty} f_N^{\text{typ}}(\beta) =\lim_{N \to \infty} \overline{f_N(\beta)} =f_\infty(\beta) }

Hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_N(\beta) } becomes effectively deterministic and its sample-to sample fluctuations vanish in relative terms:

$\lim _{N\to \infty }{\frac {\overline {f_{N}^{2}(\beta )}}{{\overline {f_{N}(\beta )}}^{2}}}=1.$

The partition function

The partition function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_N = \exp(-\beta N f_N(\beta)) }

is itself a random variable in disordered systems. Analytical methods can capture the statistical properties of this variable. We can define to average over the disorder realizations:

• The annealed average corresponds to the calculation of the moments of the partition function. The annealed free energy is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{\text{ann.}} = -\frac{1}{\beta N} \ln \, \overline{Z_N} }

• the quenched average corresponds to the average of the logarithm of the partition function, which is self-averaging for sure.

$f_{\infty }(\beta )\sim {\overline {f_{N}(\beta )}}=-\ln Z_{N}(\beta )/(\beta N)$

Do these two averages coincide?

If the partition function is self-averaging in the thermodynamic limit, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N \to \infty} Z_N(\beta) = \lim_{N \to \infty} Z_N^{\text{typ}}(\beta) =\lim_{N \to \infty} \overline{Z_N(\beta)} = e^{-\beta N f_\infty(\beta)} }

As a consequence, the annealed and the quenched averages coincide.

If the partition function is not self-averaging, only typical partition function concentrates, but extremely rare configurations contribute disproportionately to its moments:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N \to \infty} Z_N^{\text{typ}}(\beta) = e^{-\beta N f_\infty(\beta)} < \lim_{N \to \infty} \overline{Z_N(\beta)} = e^{-\beta N f^{\text{ann.}}(\beta)} }

There are then two main strategies to determine the deterministic value of the observable :

• Compute directly the quenched average using methods such as the replica trick and the Parisi solution.

• Determine the typical value $Z_{N}^{\text{typ}}(\beta )$ and evaluate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\infty(\beta) = -\frac{1}{\beta N} \ln Z_N^{\text{typ}}(\beta) }

Part II

Random Energy Model

The Random energy model (REM) neglects the correlations between the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=2^N} configurations. The energy associated to each configuration is an independent Gaussian variable with zero mean and variance $N$. The simplest solution of the model is with the microcanonical ensemble.

Microcanonical calculation

• Step 1: Number of states . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal N}_N(E) d E} the number of states of energy in the interval (E,E+dE).

It is a random number and we use the representation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{N}_N(E)dE= \sum_{\alpha=1}^{2^N} \chi_\alpha(E) dE \;}

with $\chi _{\alpha }(E)=1$ if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_\alpha \in [E, E+dE]} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_\alpha(E)=0} otherwise. We can cumpute its average

${\overline {{\mathcal {N}}_{N}(E)}}={\frac {2^{N}}{\sqrt {2\pi N}}}\exp \left(-{\frac {E^{2}}{2N}}\right)\sim \exp \left[N(\ln 2-\epsilon ^{2}/2)\right]$

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon =E/N } is the energy density and the annealed entropy density in the thermodynamic limit is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s^{\text{ann.}}(\epsilon)=\ln 2 -\epsilon^2/2 }

• Step 2: Self-averaging. Let compute now the second moment

${\overline {{\mathcal {N}}_{N}^{2}(E)}}=\sum _{\alpha =1}^{2^{N}}{\overline {\chi _{\alpha }}}\left(\sum _{\beta \neq \alpha }{\overline {\chi _{\beta }}}\right)+\sum _{\alpha =1}^{2^{N}}{\overline {\chi _{\alpha }^{2}}}\sim {\mathcal {N}}_{N}(E)\left({\mathcal {N}}_{N}(E)-\exp \left(-{\frac {E^{2}}{2N}}\right)\right)+{\mathcal {N}}_{N}(E)$

We can then check the self averaging condition:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\overline{\mathcal{N}_N^2(E)}}{\overline{\mathcal{N}_N(E)}^2} \sim 1+ \frac{1}{\overline{\mathcal{N}_N(E)}}}

A critical energy density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon^* = \sqrt{2 \ln 2}} separates two distinct regime:

• Self-averaging regime for $|\epsilon |<\epsilon ^{*}$

In this regime, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mathcal{N}_N(E)}} is exponentially large and becomes self-averaging, i.e., its value is determinstic (average, typical, median are the same). The annealed entropy density coincides with the quenched entropy density:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_\infty(\epsilon) = s^{\text{ann.}}(\epsilon) = \ln 2 - \frac{\epsilon^2}{2}}

• Non self-averaging regime for $|\epsilon |>\epsilon ^{*}$

In this regime, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mathcal{N}_N(E)}} is still exponentially small but nonzero. However, for most disorder realizations, there are no configurations with energy below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \epsilon^* N} . Only a vanishingly small fraction of rare samples gives a positive contribution to the average. As a result, the typical number of such configurations is zero, ${\mathcal {N}}_{N}^{\text{typ}}(E)=0$, and the quenched entropy becomes:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_\infty(\epsilon) = -\infty < s^{\text{ann.}}(\epsilon) = \ln 2 - \frac{\epsilon^2}{2}}

• Step 3: The partition function

The annealed partition function is the average of the partition function over the disorder:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z_N(\beta)} = \int_{-\infty}^{\infty} d\epsilon \; \overline{\mathcal{N}_N(\epsilon)} \, e^{-\beta N \epsilon}. }

Alternatively, we can focus on the typical value of the partition function, which is the value observed with probability one in the thermodynamic limit. This corresponds to the quenched average:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_N^{\text{typ}}(\beta) = \int_{-\infty}^{\infty} d\epsilon \; \mathcal{N}_N(\epsilon) \, e^{-\beta N \epsilon}. }